A Lambert quadrilateral has two sides each adjacent to two of the three right angles.

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Applied Mathematics

http://jipam.vu.edu.au/

Volume 6, Issue 4, Article 99, 2005

L’HOSPITAL-TYPE RULES FOR MONOTONICITY, AND THE LAMBERT AND SACCHERI QUADRILATERALS IN HYPERBOLIC GEOMETRY

IOSIF PINELIS

DEPARTMENT OFMATHEMATICALSCIENCES

MICHIGANTECHNOLOGICALUNIVERSITY

HOUGHTON, MICHIGAN49931 ipinelis@mtu.edu

Received 10 August, 2005; accepted 24 August, 2005 Communicated by M. Vuorinen

ABSTRACT. Elsewhere we developed rules for the monotonicity pattern of the ratiof /gof two functions on an interval of the real line based on the monotonicity pattern of the ratiof⁰/g⁰ of the derivatives. These rules are applicable even more broadly than the l’Hospital rules for limits, since we do not require that bothf andg, or either of them, tend to0or∞at an endpoint of the interval.

Here these rules are used to obtain monotonicity patterns of the ratios of the pairwise distances between the vertices of the Lambert and Saccheri quadrilaterals in the Poincaré model of hyperbolic geometry. Some of the results may seem surprising. Apparently, the methods will work for other ratios of distances in hyperbolic geometry and other Riemann geometries.

The presentation is mainly self-contained.

Key words and phrases: L’Hospital type rules for monotonicity, Hyperbolic geometry, Poincaré model, Lambert quadrilater- als, Saccheri quadrilaterals, Riemann geometry, Differential geometry.

2000 Mathematics Subject Classification. Primary 53A35, 26A48; Secondary 51M25, 51F20, 51M15, 26A24.

1. L’HOSPITAL-TYPE RULES FORMONOTONICITY

Let −∞ ≤ a < b ≤ ∞. Let f and g be differentiable functions defined on the interval (a, b), and letr := f /g. It is assumed throughout that g andg⁰ do not take on the zero value and do not change their respective signs on (a, b). In [16], general “rules” for monotonicity patterns, resembling the usual l’Hospital rules for limits, were given. In particular, according to [16, Proposition 1.9], the dependence of the monotonicity pattern ofr (on(a, b)) on that of ρ:=f⁰/g⁰(and also on the sign ofgg⁰) is given by Table 1.1, where, for instance,r &%means that there is somec∈(a, b)such thatr &(that is,ris decreasing) on(a, c)andr%on(c, b).

Now suppose that one also knows whetherr %orr &in a right neighborhood of aand in a left neighborhood ofb; then Table 1.1 uniquely determines the monotonicity pattern ofr.

ISSN (electronic): 1443-5756

239-05

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2 IOSIFPINELIS

ρ gg⁰ r

% >0 %or&or&%

& >0 %or&or%&

% <0 %or&or%&

& <0 %or&or&%

Table 1.1: Basic rules for monotonicity

Clearly, these l’Hospital-type rules for monotonicity patterns are helpful wherever the l’Hospital rules for limits are so, and even beyond that, because the monotonicity rules do not require that bothf andg(or either of them) tend to 0 or∞at any point.

The proof of these rules is very easy if one additionally assumes that the derivativesf⁰ and g⁰ are continuous and r⁰ has only finitely many roots in (a, b) (which will be the case if, for instance,r is not a constant andf andg are real-analytic functions on[a, b]): Indeed, suppose that the assumptions ρ %and gg⁰ > 0 of the first line of Table 1.1 hold. Then it suffices to show that r⁰(x) may change sign only from − to +as x increases from a to b. To obtain a contradiction, suppose the contrary, so that there is some rootuofr⁰in(a, b)such that in some right neighborhood(u, t)of the rootuone hasr⁰ < 0and hencer < r(u). Consider now the key identity

(1.1) g²r⁰ = (ρ−r)g g⁰,

which is easy to check. Then the conditionsr⁰(u) = 0andr⁰ <0on(u, t)imply, respectively, thatρ(u) = r(u)and ρ < r on (u, t). It follows that ρ < r < r(u) = ρ(u)on(u, t), which contradicts the conditionρ %. The other three lines of Table 1.1 can be treated similarly. A proof without using the additional conditions (that the derivativesf⁰ andg⁰ are continuous and r⁰ has only finitely many roots) was given in [16].

Based on Table 1.1, one can generally infer the monotonicity pattern of r given that of ρ, however complicated the latter is. In particular, one has Table 1.2.

ρ gg⁰ r

%& >0 %or&or%&or&%or&%&

&% >0 %or&or%&or&%or%&%

%& <0 %or&or%&or&%or%&%

&% <0 %or&or%&or&%or&%&

Table 1.2: Derived rules for monotonicity

In the special case when bothf andg vanish at an endpoint of the interval(a, b), l’Hospital- type rules for monotonicity and their applications can be found, in different forms and with different proofs, in [9, 11, 14, 8, 2, 3, 1, 4, 5, 15, 16, 17, 18].

The special-case rule can be stated as follows: Suppose thatf(a+) =g(a+) = 0orf(b−) = g(b−) = 0; suppose also that ρ is increasing or decreasing on the entire interval(a, b); then, respectively, ris increasing or decreasing on(a, b). When the conditionf(a+) = g(a+) = 0 orf(b−) =g(b−) = 0does hold, the special-case rule may be more convenient, because then

J. Inequal. Pure and Appl. Math., 6(4) Art. 99, 2005 http://jipam.vu.edu.au/

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one does not have to investigate the monotonicity pattern of ratio r near the endpoints of the interval(a, b).

The special-case rule is easy to prove. For instance, suppose thatf(a+) =g(a+) = 0. Then g andg⁰ must have the same sign on(a, b). By the mean-value theorem, for everyx ∈ (a, b) there is someξ ∈(a, x)such thatr(x) =ρ(ξ). Now the rule follows by identity (1.1).

This latter proof is essentially borrowed from [2, Lemma 2.2]. Another very simple proof of the special-case rule was given in [15]; that proof remains valid under somewhat more general conditions onf andg. A unified treatment of the monotonicity rules, applicable whether or not f andg vanish at an endpoint of(a, b), can be found in [16].

(L’Hospital’s rule for the limit r(b−) (say) when g(b−) = ∞ does not have a “special- case” analogue for monotonicity, even if one also has f(b−) = ∞. For example, consider f(x) = x−1−e^−xandg(x) =xforx >0. Thenr %on(0,∞), even thoughρ&on(0,∞) andf(∞−) =g(∞−) =∞.)

In view of what has been said here, it should not be surprising that a very wide variety of applications of these l’Hospital-type rules for monotonicity patterns were given: in areas of analytic inequalities [15, 16, 19, 5], approximation theory [17], differential geometry [8, 9, 11], information theory [15, 16], (quasi)conformal mappings [1, 2, 3, 4], statistics and probability [14, 16, 17, 18], etc.

Clearly, the stated rules for monotonicity could be helpful whenf⁰org⁰can be expressed sim- pler thanf org, respectively. Such functionsf andg are essentially the same as the functions that could be taken to play the role ofuin the integration-by-parts formulaR

u dv=uv−R v du;

this class of functions includes polynomial, logarithmic, inverse trigonometric and inverse hyperbolic functions, and as well as non-elementary “anti-derivative” functions of the form x7→Rx

a h(u)duorx7→Rb

xh(u)du.

(“Discrete” analogues, forfandgdefined onZ, of the l’Hospital-type rules for monotonicity, are available as well [20].)

In the present paper, we use the stated rules for monotonicity to obtain monotonicity properties of the Lambert and Saccheri quadrilaterals in hyperbolic geometry. This case represents a perfect match between the two areas. Indeed, the distances in hyperbolic geometry are expressed in terms of inverse hyperbolic functions, whose derivatives are algebraic. One can expect these rules to work for other Riemann geometries as well, since the geodesic distances there are line integrals, too.

2. MONOTONICITYPROPERTIES OF THELAMBERT AND SACCHERI

QUADRILATERALS

2.1. Background.

2.1.1. Hyperbolic plane. The Lambert and Saccheri quadrilaterals are quadrilaterals in the Poincaré hyperbolic planeH².

The significance of the Poincaré model is that, by the Riemann mapping theorem, any simply connected analytic Riemann surface is conformally equivalent toH²,C, or C∪ {∞}[7, The- orem 9.1]. Moreover, any analytic Riemann surface is conformally equivalent to the quotient surfaceR/G, where˜ R˜ isH²,C, orC∪ {∞}, andGis a group of Möbius transformations act- ing discontinuously on (the covering surface)R˜[7, Proposition 9.2.3]. However, this comment will not be used further in this paper.

To make this section mainly self-contained, let us fix the terminology and basic facts concerning the Poincaré model of hyperbolic plane geometry. The set of points in this model is the upper half-plane

H² :={z ∈C: Imz >0}.

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4 IOSIFPINELIS

This set is endowed with the differential metric element ds:= |dz|

Imz,

so that the length of any rectifiable curve inH²is obtained as the line integral ofds. Forx∈R andr ∈R\ {0}, let us refer to the semicircles

[[x−r, x+r]] := {z∈H²: |z−x|=|r|}, centered at pointxand of radius|r|, and the vertical rays

[[x,∞]] :={z ∈H²: Rez=x}

as the “lines”. It will be seen in a moment that these “lines” are precisely the geodesics in this geometry, so that the geodesics are orthogonal to the real axis.

Forx∈Randr ∈R\ {0}, letι_x,rdenote the reflection ofH²in the semicircle[[x−r, x+r]], so that, forz ∈H²,

ι_x,r(z) :=x+ r² z−x.

It is easy to see that this transformation is inverse to itself and preserves H² as well as the metric element ds, and hence also the (absolute value of the) angles. Indeed, ifw := ι_x,r(z) forz ∈H², then Imw= r² Imz/|z−x|² anddw =−r²dz/(z−x)², so that Imw > 0and

|dw|/Imw=|dz|/Imz.

Let G be the group of transformations of H² generated by all such reflections. Then G preserves the metric element ds. Note that G contains all the homotheties z 7→ η_x,λ(z) :=

x +λ(z − x), horizontal parallel translations z 7→ σ_x(z) := z + x, and reflections z 7→

ιx,∞(z) := 2x−z in the vertical rays[[x,∞]], wherex∈Randλ >0; indeed,η_x,λ=ι_x,^√_λ◦ι_x,1, ιx,∞=ι_x+r,2r◦ιx−r,2r◦ι_x+r,2r, andσ_x =ιx/2,∞◦ι0,∞.

It is easy to see that the geodesic connecting two points z₁ andz₂ on the same vertical ray [[x,∞]](x ∈ R) is the segment of that ray with the endpoints z1 andz2, so that the geodesic distance d(z1, z2) between suchz1 and z2 is|ln(y1/y2)|, whereyj := Imzj, j = 1,2. Now it is seen that group Gacts transitively on the set of all ordered pairs(z₁, z₂)of points on the vertical ray [[x,∞]] with a fixed value of the distance d(z₁, z₂) — in the sense that, for any two pairs (z₁, z₂)and (w₁, w₂)of points on [[x,∞]] withd(z₁, z₂) = d(w₁, w₂), there is some transformationg inGsuch thatg(z_j) =w_j,j = 1,2; indeed, it suffices to takeg to be a single reflectionι_x,r or a single homothetyη_x,λ, for somer >0orλ >0.

Next, the reflectionι_x+r,2rmaps the semicircle[[x−r, x+r]]onto the vertical ray[[x−r,∞]], and hence vice versa, for allx ∈ Randr ∈ R\ {0}. Moreover, any two distinct points inH² lie on exactly one “line”.

It follows now that indeed the “lines” are precisely the geodesics, and group Gacts transitively on the set of all ordered pairs(z₁, z₂)of points inH²with any fixed value of the geodesic distanced(z₁, z₂). Another corollary here is the formula for the geodesic distance between any two pointsz1andz2ofH²:

(2.1) d(z₁, z₂) = arcch

1 + |z₁−z₂|² 2 Imz₁ Imz₂

, wherearcchx := ln x+√

x²−1

forx > 1; cf. [6, Theorem 7.2.1(ii)]. One can now also easily derive Pythagoras’ theorem,

(2.2) chc= cha chb,

for a right-angled (geodesic) triangle ABC with side c opposite to the right-angle vertex C and two other sides a andb; indeed, such a triangle is G-congruent, for some k ∈ (0,1)and

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θ ∈ (0, π/2), to the triangle with vertices C∗ = i, A∗ = k i, and B∗ = e^iθ; cf. [6, Theorem 7.11.1]. (Yet another corollary, not to be used in this paper, is thatGis the group of all isometries ofH².)

2.1.2. Lambert’s and Saccheri’s quadrilaterals. A Lambert quadrilateral is a quadrilateral in the Poincaré hyperbolic plane with anglesπ/2,π/2,π/2, andϕ, for someϕ; a Saccheri quadrilateral is a quadrilateral (also in the hyperbolic plane) with anglesπ/2,π/2,ψ andψ, for some ψ [6, Section 7.17]. See Figure 2.1.

For a Saccheri quadrilateral, let us refer to (the length of) its side adjacent to the right angles as the base, its opposite side as the top, and to either of the other two (congruent to each other) sides simply as the side.

A Lambert quadrilateral has two sides each adjacent to two of the three right angles. Let us arbitrarily choose one of these two sides and refer to it as the base, and to the other one of the two as the (short) side. The side opposite to the base will again be referred to as the top, and the fourth side as the long side. It will be seen in the next subsection that indeed the long side is always longer than the short one.

LAMBERT’S AND SACCHERI’S QUADRILATERALS 5

2.1.2. Lambert’s and Saccheri’s quadrilaterals. A Lambert quadrilateral is a quadrilateral in the Poincar´ e hyperbolic plane with angles π/2, π/2, π/2, and ϕ, for some ϕ; a Saccheri quadrilateral is a quadrilateral (also in the hyperbolic plane) with angles π/2, π/2, ψ and ψ, for some ψ [6, Section 7.17]. See Figure 2.1.

For a Saccheri quadrilateral, let us refer to (the length of) its side adjacent to the right angles as the base, its opposite side as the top, and to either of the other two (congruent to each other) sides simply as the side.

A Lambert quadrilateral has two sides each adjacent to two of the three right angles.

Let us arbitrarily choose one of these two sides and refer to it as the base, and to the other one of the two as the (short) side. The side opposite to the base will again be referred to as the top, and the fourth side as the long side. It will be seen in the next subsection that indeed the long side is always longer than the short one.

A

_L

D

_L

A

_S

D

_S

B C

Figure 2.1: Lambert’s (A_LBCD_L) and Saccheri’s (A_SBCD_S) quadrilaterals; A_LB, A_LD_L, BC, and CD_L are respectively the base, short side, long side, and top of the Lambert quadrilateral; A_SB, A_SD_S = BC, andCD_S are respectively the base, side, and top of the Saccheri quadrilateral; the angles at vertices A_S, B, A_L, and D_L are π/2.

It follows from the discussion in Subsubsection 2.1.1 that the group G acts transitively on the set of all Saccheri quadrilaterals with any given values of the base and the side, as well as on the set of all Lambert quadrilaterals with any given values of the base and the short side. That is, all Saccheri quadrilaterals with any given values of the base and the side are G-congruent to each other, and so, they have the same geodesic distances between any two of their corresponding vertices. The same holds for all Lambert quadrilaterals with any given values of the base and the short side.

2.2. Main results.

2.2.1. Lambert quadrilaterals. In view of the conclusions of Subsection 2.1, any Lambert quadrilateral is G-congruent, for some

k ∈ (0, 1) and θ ∈ (0, π/2), to the particular Lambert quadrilateral ABCD with vertices

A = k i, B = i, C = e

^iθ

, D = k e

^iψ

, where ψ := arccos

ch(ln k) cos θ

Figure 2.1: Lambert’s (A_LBCD_L) and Saccheri’s (A_SBCD_S) quadrilaterals;A_LB,A_LD_L,BC, andCD_Lare respectively the base, short side, long side, and top of the Lambert quadrilateral;A_SB,A_SD_S =BC, andCD_S are respectively the base, side, and top of the Saccheri quadrilateral; the angles at verticesA_S,B,A_L, andD_L areπ/2.

It follows from the discussion in Subsubsection 2.1.1 that the groupGacts transitively on the set of all Saccheri quadrilaterals with any given values of the base and the side, as well as on the set of all Lambert quadrilaterals with any given values of the base and the short side. That is, all Saccheri quadrilaterals with any given values of the base and the side areG-congruent to each other, and so, they have the same geodesic distances between any two of their corresponding vertices. The same holds for all Lambert quadrilaterals with any given values of the base and the short side.

2.2. Main Results.

2.2.1. Lambert quadrilaterals. In view of the conclusions of Subsection 2.1, any Lambert quadrilateral isG-congruent, for some

k ∈(0,1) and θ ∈(0, π/2),

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6 IOSIFPINELIS

to the particular Lambert quadrilateralABCDwith vertices

A = k i, B = i, C = e^iθ, D = k e^iψ, where ψ := arccos ch(lnk) cosθ (see Figure 2.2), so that, by (2.1),

AB= ln 1

k, BC = arcchc, CD = arcch1 +k² q , , AD= arcch2c k

q , AC = arcchc(1 +k²)

2k , BD = arcchc(1 +k²)

q ,

(2.3)

where q :=p

(1 +k²)²−c²(1−k²)² and c:= 1/sinθ.

(2.4)

(One can verify, using (2.2) and (2.3), that indeed∠A = ∠B = ∠C = π/2.) Then one may refer toABas the base, of lengthln(1/k), and toBC as the short side, of lengtharcchc. Note that, for the pointDto exist inH², one must havech(lnk) cosθ <1, which is equivalent to

1< c < c_k, where c_k := 1 +k² 1−k².

Let us fix (the length of) the base AB (so that k ∈ (0,1) is fixed) and letc increase from 1 to c_k, so that the short side BC = arcchc increases from 0 to arcchc_k. The goal here is to determine the monotonicity patterns of ⁶₂

= 15 completely representative pairwise ratios r = CD/AD, CD/BD, . . . , BC/AB of the ⁴₂

= 6(geodesic) distances between the four verticesA,B,C,D. For each pair of such distances, it is enough to consider only one of the two mutually reciprocal ratios; indeed, for example, the monotonicity pattern of the ratioCD/AD determines that ofAD/CD. All the ratios r will be expressed as functions of c. (We do not distinguish in terminology or notation between a segment of a geodesic and its length.)

(see Figure 2.2), so that, by (2.1),

AB = ln 1

k , BC = arcch c, CD = arcch 1 + k

²

q , , AD = arcch 2 c k

q , AC = arcch c (1 + k

²

)

2 k , BD = arcch c (1 + k

²

) q , (2.3)

where q :=

(1 + k

²

)

²

− c

²

(1 − k

²

)

²

and c := 1/ sin θ.

(2.4)

(One can verify, using (2.2) and (2.3), that indeed ∠ A = ∠ B = ∠ C = π/2.) Then one may refer to AB as the base, of length ln(1/k ), and to BC as the short side, of length arcch c. Note that, for the point D to exist in H

²

, one must have ch(ln k) cos θ < 1, which is equivalent to

1 < c < c

_k

, where c

_k

:= 1 + k

²

1 − k

²

.

Let us ﬁx (the length of) the base AB (so that k ∈ (0, 1) is ﬁxed) and let c increase from 1 to c

_k

, so that the short side BC = arcch c increases from 0 to arcch c

_k

. The goal here is to determine the monotonicity patterns of

₆

2

= 15 completely representative pairwise ratios r = CD/AD, CD/BD, . . . , BC/AB of the

₄

2

= 6 (geodesic) distances between the four vertices A, B, C, D. For each pair of such distances, it is enough to consider only one of the two mutually reciprocal ratios; indeed, for example, the monotonicity pattern of the ratio CD/AD determines that of AD/CD. All the ratios r will be expressed as functions of c. (We do not distinguish in terminology or notation between a segment of geodesics and its length.)

to p short

side

long sid e

sh ort diago nal

base

A B

C

D

Figure 2.2: A Lambert quadrilateral: ∠A= ∠B =∠C =π/2

Theorem 2.1. The monotonicity patterns of the 15 representative ratios r(c) are given by Table 2.1, where k

_∗

:= √

2 − 1.

One simple corollary here is that, of the two sides (BC and AD) of the Lambert quadrilateral, BC is indeed always the shorter one (this is obvious from Figure 2.2 as well). Also, of the two diagonals (AC and BD) of the quadrilateral, AC is always the shorter one.

What is perhaps surprising is that the monotonicity patterns of two ratios, CD/AC (top-to-short-diagonal) and CD/AD top-to-long-side), turn out to depend on (the ﬁxed

Figure 2.2: A Lambert quadrilateral:∠A=∠B=∠C=π/2

Theorem 2.1. The monotonicity patterns of the 15 representative ratiosr(c)are given by Ta- ble 2.1, wherek∗ :=√

2−1.

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r Pattern for eachkin

r(1+) r(c_k−) Comments (0,1) (0, k∗] (k∗,1)

CD/AC % &% 1 ∞

CD/AD & &% ∞ 1

CD/BC &% ∞ ∞ ∃c∈(1, c_k)r(c) = 1

⇐⇒ k>1/√ 3

CD/BD &% 1 1

CD/AB % 1 ∞

AC/AD & ∞ 0

AC/BC & ∞ >1

AC/BD & 1 0

AC/AB % 1 >1

BD/AD & ∞ 1

BD/BC &% ∞ ∞ ∀k∈(0,1)∀c∈(1, c_k) r(c)>1

BD/AB % 1 ∞

AD/BC % >1 ∞

AD/AB % 0 ∞

BC/AB % 0 r(c_k−) r(c_k−)>1 ⇐⇒ k > k∗

Table 2.1: Monotonicity patterns for the ratios in the Lambert quadrilateral

One simple corollary here is that, of the two sides (BCandAD) of the Lambert quadrilateral, BC is indeed always the shorter one (this is obvious from Figure 2.2 as well). Also, of the two diagonals (AC andBD) of the quadrilateral,AC is always the shorter one.

What is perhaps surprising is that the monotonicity patterns of two ratios, CD/AC (top- to-short-diagonal) andCD/AD (top-to-long-side), turn out to depend on (the fixed length of) the baseAB = ln(1/k) of the quadrilateral. When the base AB is smaller than ln(1/k∗) = ln(1 +√

2), these two ratios are not monotonic.

Three other ratios — CD/BC (top-to-short-side), CD/BD (top-to-long-diagonal), and BD/BC (long-diagonal-to-short-side) — are not monotonic for any given base; however, this should not be surprising, since for each of these three ratiosrone hasr(1+) =r(c_k−).

In particular, it follows that of all the 5 ratios of the top to the other lengths, only the trivial one, the ratioCD/ABof the top to the fixed base, is monotonic for every given base.

Another small-base peculiarity shows up for two ratios, CD/BC (top-to-short-side) and BC/AB(short-side-to-base); namely, these ratios take on values to both sides of1iff the base is small enough – smaller than ln√

3 in the case of CD/BC and smaller than ln(1/k∗) = ln(1 +√

2)in the case ofBC/AB.

Proof of Theorem 2.1. From (2.3), it is clear that the 5 ratios of BC, CD, AD, AC, and BD to the fixed AB are increasing (in c), and the inequality BC/AB > 1 can be rewritten as chBC > chAB, which is equivalent to k > k∗. The monotonicity pattern for AC/AD =

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8 IOSIFPINELIS

(AC/BD)(BD/AD) obviously follows from those for AC/BD andBD/AD. It remains to consider the other 9 of the 15 ratios.

In terms of the expressionq, defined by (2.4), and the expressions q₁ :=p

(c²−1)(1 +k²)²+ (1−k²)², q₂ :=√

c²−1, (2.5)

q3 :=p

2(c²−1)(1 +k⁴) + (1−k²)², (2.6)

one computes the ratios,ρ, of the derivatives of the distances with respect toc:

(CD)⁰

(AC)⁰ = (1−k²)q₁

q² , (CD)⁰

(AD)⁰ = (1−k²)q₂

2k , (CD)⁰

(BD)⁰ = (1−k²)q₃ (1 +k²)² , (AC)⁰

(BC)⁰ = (1 +k²)q₂

q₁ , (AC)⁰

(BD)⁰ = q²q₃

(1 +k²)²q₁, (BD)⁰

(AD)⁰ = (1 +k²)²q₂ 2k q₃ , (AD)⁰

(BC)⁰ = 2k(1 +k²)

q² , (CD)⁰

(BC)⁰ = (CD)⁰ (AC)⁰

(AC)⁰

(BC)⁰, (BD)⁰

(BC)⁰ = (BD)⁰ (AD)⁰

(AD)⁰ (BC)⁰.

Of these 9 ratios, it is now clear that 8 ratios (except (AC)⁰/(BD)⁰) are increasing (in c). Hence, by the first line of Table 1.1, each of the corresponding 8 ratios, r, of distances, CD/AC, . . . , AD/BC (except forAC/BD), has one of these three patterns: %, &, or&%.

(It can be shown that(AC)⁰/(BD)⁰ is&or%&, depending on whether the base,AB, is large enough; however, this fact will not be used in this paper.)

Now let us consider each of the 8 “unexceptional” ratios separately, after which the “excep- tional” ratio,AC/BD, will be considered.

(1) r(c) = CD/AC: Here it is obvious thatr(1+) = 1andr(ck−) = ∞. This excludes the patternr &. To discriminate between the possibilitiesr &andr &%, it suffices to determine whether there exists some c∈ (1, c_k)such thatr(c) = 1or, equivalently, chCD = chAC. Now it is easy to complete the proof of Theorem 2.1 for the ratio r(c) = CD/AC.

(2) r(c) = CD/AD: Here it is obvious thatr(1+) = ∞. By l’Hospital’s rule for limits, r(c_k−) =ρ(c_k−) = 1. This excludes the patternr %. Moreover, it is easy to see, as in the previous case, that there exists somec∈(1, c_k)such thatr(c) = 1iffk > k∗. (3) r(c) = CD/BC: Herer(1+) =r(c_k−) =∞. Hence,r&%. Moreover, it is easy to

see that there exists somec∈(1, c_k)such thatr(c) = 1iffk >1/√ 3.

(4) r(c) = CD/BD: Herer(1+) = 1. By l’Hospital’s rule for limits,r(c_k−) =ρ(c_k−) = 1. Hence,r&%.

(5) r(c) =AC/BC: Here, withµ:= 2k(1 +k²)andν :=√

1 + 14k⁴+k⁸, one has the following atc=c_k−:

r⁰· 2k ν BC²

(1−k²)AC =µBC

AC −ν < µ−ν,

since, in view of (2.3),BC < AC. Butµ²−ν² =−(1−k²)⁴ <0. Hence,r⁰(c_k−)<0, so thatr&in a left neighborhood ofc_k. Thus,r&.

(6) r(c) = BD/AD: Here r(1+) = ∞. By l’Hospital’s rule for limits, r(c_k−) = ρ(c_k−) = 1. In view of (2.3), herer > 1on(1, c_k). Hence, r is decreasing on(1, c_k) from∞to1.

(7) r(c) = BD/BC: Here r(1+) = r(c_k−) = ∞. Hence, r &%on (1, c_k). Also, in view of (2.3), one has herer >1on(1, c_k).

(8) r(c) = AD/BC: Here, by the special-case rule for monotonicity,r %. By l’Hospital’s rule,r(1+) =ρ(1+) = (1 +k²)/(2k)>1. Also, it is obvious thatr(c_k−) =∞.

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It remains to consider the 9th ratio,

¶ r(c) =AC/BD: Here, as was stated,ρ(c) := (AC)⁰/(BD)⁰is non-monotonic incfor k in a left neighborhhood of 1. This makes it more difficult to act as in the cases considered above, since the rootcof the equationρ⁰(c) = 0depends onk. However, what helps here is that the monotonicity pattern ofrturns out to be simple, as will be proved in a moment: r &. One can use the following lemma, whose proof is based on the special-case rule for monotonicity stated in Section 1.

Lemma 2.2. Forx>1, let λ(x) :=

√x²−1 arcchx

x³ , α(x) := x²−1

x³ , β(x) :=

√x²−1 x³ . Then for alluandv in(1,∞)

λ(v)

λ(u) 6max

α(v) α(u),β(v)

β(u)

.

Proof of Lemma 2.2. Obviously,λ/β = arcch %. Hence, ^λ(v)_λ(u) 6 _β(u)^β(v) if1< v 6 u. It remains to consider the case when1< u < v. Note that

(arcchx)⁰

√x²−1⁰ = 1 x

is decreasing inx >1. Hence, by the special-case rule for monotonicity, λ(x)

α(x) = arcchx

√x²−1

is decreasing inx >1. Hence, ^λ(v)_λ(u) < ^α(v)_α(u) if1< u < v.

Let us now return to the consideration of the ratior(c) = AC/BD. It suffices to show that r⁰(c)<0for allk ∈(0,1)andc∈(1, c_k). One has the identity

r⁰(c)2BD²k√

u²−1√ v²−1

(1 +k²)λ(u)v³ = λ(v) λ(u) −K, where

u:= c(1 +k²)

2k , v := c(1 +k²) q

(1 +k²)²−c²(1−k²)²

, K :=

1 +k² 2k

2.

Therefore and in view of Lemma 2.2, it suffices to show that the expressions P :=

α(v) α(u)

2

−K²

!

α(u)² 4c⁶k²(1 +k²)⁶ (1−k²)² and Q:=

β(v) β(u)

2

−K²

!

β(u)²c⁶(1 +k²)⁶ (1−k²)²

are negative for allk ∈(0,1)andc∈(1, c_k). But this can be done in a completely algorithmic manner, sinceP andQare polynomials inkandc, andc_kis a rational function ofk[21, 12, 10].

With Mathematica, one can use the command Reduce[P>=0 && 1<c<ck && 0<k<1]

(whereckstands forc_k), which outputsFalse, meaning that indeedP < 0for allk ∈(0,1) andc∈(1, c_k); similarly, forQin place ofP.

Theorem 2.1 is proved.

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10 IOSIFPINELIS

2.2.2. Saccheri quadrilaterals. LetABCDbe a Saccheri quadrilateral. Here one may assume that

A =k i, B =i, C =e^iθ, D=k e^iθ,

where again 0 < k < 1and 0 < θ < π/2, so that the angles at verticesA and B are right, and BC = AD, so that BD = AC. Let us refer here toAB = ln(1/k) as the base and to BC =AD= arcchc as the side, where againc:= 1/sinθ.Herecvaries from1to∞.

2.2.2. Saccheri quadrilaterals. Let ABCD be a Saccheri quadrilateral. Here one may assume that

A = k i, B = i, C = e

^iθ

, D = k e

^iθ

,

where again 0 < k < 1 and 0 < θ < π/2, so that the angles at vertices A and B are right, and BC = AD, so that BD = AC . Let us refer here to AB = ln(1/k) as the base and to BC = AD = arcch c as the side, where again c := 1/ sin θ. Here c varies from 1 to ∞ .

top sid e

sid e

base

A B

D C

Figure 2.3: A Saccheri quadrilateral: ∠A= ∠B = π/2 and ∠C = ∠D, whence AD = BC and AC = BD

Again, let us ﬁx the base AB = ln(1/k) (so that k ∈ (0, 1) is ﬁxed); also, let c increase from 1 to ∞ , so that the side BC = AD = arcch c increases from 0 to ∞ . Here, taking into account the equalities BC = AD and BD = AC, we have to determine the monotonicity patterns of

₄

2

= 6 completely representative pairwise ratios.

Theorem 2.3. The monotonicity patterns of the 6 ratios r(c) are given by Table 2.2.

r Pattern for each k in

r(1+) r( ∞− ) k

_∗∗

(0, 1) (0, k

_∗∗

] (k

_∗∗

, 1)

CD/AD ∞ 2 k

_∗²

= 3 − 2 √

2 CD/BD 1 2 2 − √

3 CD/AB 1 ∞

AD/BD 0 1

AD/AB 0 ∞

BD/AB 1 ∞

Table 2.2: Monotonicity patterns for the ratios in the Saccheri quadrilateral

Thus, the diagonal AC = BD always exceeds both the base AB and the side AD = BC . Also, the top CD always exceeds the base.

Recently it was observed by Pambuccian [13] that the ratio CD/BD = CD/AC of the top of a Saccheri quadrilateral to its diagonal may be less than or greater than or equal to

Figure 2.3: A Saccheri quadrilateral:∠A=∠B =π/2and∠C=∠D, whenceAD=BCandAC=BD

Again, let us fix the baseAB = ln(1/k)(so thatk∈(0,1)is fixed); also, letcincrease from 1to∞, so that the sideBC =AD= arcchcincreases from0to∞. Here, taking into account the equalitiesBC = AD andBD = AC, we have to determine the monotonicity patterns of

4 2

= 6completely representative pairwise ratios.

Theorem 2.3. The monotonicity patterns of the 6 ratiosr(c)are given by Table 2.2.

Thus, the diagonal AC = BD always exceeds both the baseAB and the sideAD = BC.

Also, the topCDalways exceeds the base.

Recently it was observed by Pambuccian [13] that the ratio CD/BD = CD/AC of the top of a Saccheri quadrilateral to its diagonal may be less than or greater than or equal to 1.

The second line of Table 2.2 provides more information in that respect. In particular, one can see now that the top-to-diagonal ratio can be less than 1 only if the base AB is smaller than ln(2 +√

3). On the other hand, this ratio is always less than2.

Similarly to the case of the Lambert quadrilateral, the monotonicity patterns of two ratios, CD/AD (top-to-side) and CD/BD (top-to-diagonal), turn out to depend on the base AB= ln(1/k)of the quadrilateral. When the base is smaller than the threshold valueln(1/k∗∗), these two ratios are not monotonic. However, in contrast with Lambert quadrilaterals, here the threshold values for these two ratios are different from each other. Yet, for Saccheri quadrilaterals as well, it is the small base values that may result in non-monotonic patterns.

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r Pattern for eachk in

r(1+) r(∞−) k∗∗

(0,1) (0, k∗∗] (k∗∗,1)

CD/AD & &% ∞ 2 k²_∗ = 3−2√ 2

CD/BD % &% 1 2 2−√

3

CD/AB % 1 ∞

AD/BD % 0 1

AD/AB % 0 ∞

BD/AB % 1 ∞

Table 2.2: Monotonicity patterns for the ratios in the Saccheri quadrilateral

Proof of Theorem 2.3. In view of (2.1), here one has

(2.7)

AB = ln1

k, AD=BC = arcchc, CD= arcchc²(1−k)²+ 2k

2k ,

AC =BD= arcchc(1 +k²) 2k .

From these expressions, the statements of Theorem 2.3 concerning the three ratios of the top (CD), side (AD = AC), and diagonal (AC = BD) to the fixed base (AB) are obvious. It remains to consider the other three ratios.

¶ r(c) = CD/AD: This case follows immediately from the case of the top-to-long-side ratio for the Lambert quadrilateral, which latter is a “half” of a Saccheri one; see Figure 2.1.

Indeed, if the side of a Saccheri quadrilateral equals the long side of a Lambert quadrilateral and the base of the Saccheri quadrilateral is twice the base of the Lambert quadrilateral, then the top of the Saccheri quadrilateral is twice the top of the Lambert quadrilateral.

¶ r(c) = CD/BD: Here (recall (2.5))ρ(c) = 2 (1−k)q₁/((1 +k²)q₄), whereq₄ :=

p(c²−1)(1−k)²+ (1 +k)². Hence, ρ %, and so, r % or r & or r %&. Obviously, r(1+) = 1. By l’Hospital’s rule, r(∞−) = ρ(∞−) = 2. Moreover, it is easy to see that (∃

c >1r(c) = 1) iff2−√

3< k <1. This proves the second line of Table 2.2.

¶ r(c) = AD/BD: Hereρ(c) = q₁/((1 +k²)q₂), so thatρ&. Obviously,r(1+) = 0.

By l’Hospital’s rule,r(∞−) =ρ(∞−) = 1. Also, (2.7) impliesr <1. It follows thatr%.

Theorem 2.3 is proved.

2.3. Conclusion. It seems quite likely that one could similarly examine the monotonicity pat- terns of these ratios for the Lambert and Saccheri quadrilaterals under conditions other than that of a fixed base. Likewise, one could examine the monotonicity patterns of other ratios of distances, in this or other Riemann geometries.

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